Step: 1

42x ^{2} - 71x + 30

[Given trinomial.]

Step: 2

[Compare with a x ² + b x + c .]

Step: 3

Find two numbers whose product is 42 and another two numbers whose product is 30.

Factors of 42 Factors of 30

1, 42 and 6, 7 - 5, - 6

Factors of 42 Factors of 30

1, 42 and 6, 7 - 5, - 6

Step: 4

Use FOIL method to check the middle term in the trial factors.

Trial factors Middle term

(x - 5)(42x - 6) - 6x - 210x = - 216x

(6x - 5)(7x - 6) - 36x - 35x = - 71x

Trial factors Middle term

(

(6

Step: 5

So, 42x ^{2} - 71x + 30 = (6x - 5)(7x - 6).

Correct Answer is : (6x - 5)(7x - 6)

Step: 1

3y ^{2} - 10y - 8

[Given trinomial.]

Step: 2

[Compare with ax ² + bx + c .]

Step: 3

Find two numbers whose product is 3 and another two numbers whose product is - 8.

Factors of 3 Factors of - 8

1, 3 - 1, 8; - 2, 4 and - 4, 2

Factors of 3 Factors of - 8

1, 3 - 1, 8; - 2, 4 and - 4, 2

Step: 4

Use FOIL method to check the middle term in the trial factors.

Trial factors Middle term

(y - 1)(3y + 8) 8y - 3y = 5y

(y - 2)(3y + 4) 4y - 6y = - 2y

(y - 4)(3y + 2) 2y - 12y = - 10y

Trial factors Middle term

(

(

(

Step: 5

So, 3y ^{2} - 10y - 8 = (y - 4)(3y + 2).

Correct Answer is : (y - 4)(3y + 2)

Step: 1

3x ^{2} - 10x y - 25y ^{2}

[Given trinomial.]

Step: 2

Step: 3

Find two numbers whose product is 3 and another two numbers whose product is - 25.

Factors of 3 Factors of - 25

1, 3 1, - 25 and 5, - 5

Factors of 3 Factors of - 25

1, 3 1, - 25 and 5, - 5

Step: 4

Use FOIL method to check the middle term in the trial factors.

Trial factors Middle term

(x + y )(3x - 25y ) - 25xy + 3xy = - 22xy

(x + 5y )(3x - 5y ) - 5xy + 15xy = 10xy

(3x + 5y )(x - 5y ) - 15xy + 5xy = - 10xy

Trial factors Middle term

(

(

(3

Step: 5

So, 3x ^{2} - 10x y - 25y ^{2} = (3x + 5y )(x - 5y ).

Correct Answer is : (3x + 5y )(x - 5y )

Step: 1

Step: 2

Step: 3

1, - 10 | - 9 | - 10 | |

- 2, 5 | 3 | - 10 |

[Select the values of p and q by trial and error.]

Step: 4

The required values of p and q are - 2, 5.

[p + q = 3, p q = - 10.]

Step: 5

Therefore, x 2 + 3 x - 1 0 = (x - 2)(x + 5)

[Substitute for p and q .]

Correct Answer is : (x - 2)(x + 5)

Step: 1

Step: 2

Step: 3

1, 40 | 41 | 40 | |

20, 2 | 22 | 40 | |

8, 5 | 13 | 40 |

[Select the values of p and q by trial and error.]

Step: 4

The required values of p and q are 8, 5.

[p + q = 13, p q = 40.]

Step: 5

Therefore, x 2 + 1 3 x + 4 0 = (x + 8)(x + 5).

[Substitute for p and q .]

Correct Answer is : (x + 8)(x + 5)

Step: 1

Step: 2

Step: 3

1, - 70 | - 69 | - 70 | |

- 10, 7 | - 3 | - 70 |

[Select the values of p and q by trial and error.]

Step: 4

The required values of p and q are - 10, 7.

[p + q = - 3, p q = - 70.]

Step: 5

Therefore, y 2 - 3 y - 7 0 = (y - 10)(y + 7).

[Substitute for p and q .]

Correct Answer is : (y - 10)(y + 7)

Step: 1

Let the width of the ground = x ft.

Step: 2

Length of the ground = (x + 6) ft.

Step: 3

Area of the ground = 55 sq.ft.

Step: 4

So, x (x + 6) = 55

[Area of a rectangle = length × width.]

Step: 5

Step: 6

Step: 7

The factors of a trinomial x 2 + bx + c are in the form (x + p )(x + q ), where b = p + q and c = pq .

Step: 8

Compare the left side of the equation with x 2 + bx + c to get b and c values. So, b = 6 and c = - 55.

Step: 9

Find the numbers p and q whose product is - 55 and whose sum is 6.

Step: 10

- 11, 5 | - 6 | - 55 | |

11, - 5 | 6 | - 55 |

[Select the values of p and q by trial and error.]

Step: 11

The required values of p and q are 11, - 5.

[p + q = 6 , pq = - 55.]

Step: 12

[Substitute for p and q .]

Step: 13

So, the equation x 2 + 6 x - 5 5 = 0 can be written as (x + 11)(x - 5) = 0

Step: 14

Step: 15

Step: 16

The width of the ground = 5 ft.

[Negative values for dimension do not make sense.]

Step: 17

The length of the ground = 5 + 6 = 11 ft.

Correct Answer is : 11 ft

Step: 1

The factors of a trinomial x ^{2} + bx + c are in the form (x + p )(x + q ), where b = p + q and c = pq .

Step: 2

Compare the equation with x ^{2} + bx + c to get b and c values. So, b = 6 and c = 9.

Step: 3

Find the numbers p and q whose product is 9 and sum is 6.

Step: 4

1, 9 10

3, 3 6

Step: 5

The required values of p and q are 3 and 3.

Step: 6

So, x ^{2} + 6x + 9 = (x + 3)(x + 3).

Correct Answer is : (x + 3)(x + 3)

Step: 1

Step: 2

Step: 3

- 1, - 3 | - 4 | 3 |

[Select the values of p and q by trial and error.]

Step: 4

The required values of p and q are - 1 and - 3.

[p + q = - 4, pq = 3.]

Step: 5

So, x ^{2} - 4x + 3 = (x - 1)(x - 3).

[Substitute for p and q .]

Correct Answer is : (x - 1)(x - 3)

Step: 1

The factors of a trinomial x ^{2} + bx + c are in the form (x + p )(x + q ), where b = p + q and c = pq .

Step: 2

Compare the equation with x ^{2} + bx + c to get b and c values. So, b = 2 and c = -15.

Step: 3

Since c is negative, find the numbers p and q with different signs, whose product is -15 and sum is 2.

Step: 4

-1, 15 14

-3, 5 2

Step: 5

The required values of p and q are -3 and 5.

Step: 6

So, x ^{2} + 2x - 15 = (x - 3)(x + 5).

Correct Answer is : (x - 3)(x + 5)

Step: 1

Step: 2

Step: 3

- 3, 7 | 4 | - 21 | |

- 7, 3 | - 4 | - 21 |

[Select the values of p and q by trial and error.]

Step: 4

The required values of p and q are - 7 and 3.

[p + q = - 4, pq = - 21.]

Step: 5

So, x ^{2} - 4x - 21 =(x - 7) (x + 3).

[Substitute for p and q .]

Correct Answer is : (x - 7)(x + 3)

Step: 1

The factors of a trinomial x ^{2} + bx + c are in the form (x + p )(x + q ), where b = p + q and c = pq .

Step: 2

Compare the equation with x ^{2} + bx + c to get b and c values. So, b = -3 and c = -28.

Step: 3

Since c is negative, find the numbers p and q with different signs, whose product is -28 and sum is -3.

Step: 4

- 4, 7 3

- 7, 4 - 3

Step: 5

The required values of p and q are -7 and 4.

Step: 6

So, x ^{2} - 3x - 28 = (x - 7)(x + 4).

Correct Answer is : (x - 7)(x + 4)

Step: 1

Step: 2

Compare the equation with x ^{2} + bx + c to get b and c values. So, b = -7 and c = 10.

Step: 3

Since c is positive, find the numbers p and q with the same sign, whose product is 10 and sum is -7.

Step: 4

-5, -2 -7

Step: 5

The required values of p and q are -5 and -2.

Step: 6

So, x ^{2} - 7x + 10 = (x - 5)(x - 2).

Correct Answer is : (x - 5)(x - 2)

Step: 1

Step: 2

Step: 3

Step: 4

1, 6 7 6

Step: 5

2, 3 5 6

Step: 6

The required values of p and q are 2 and 3.

Step: 7

Therefore, x 2 + 5 x + 6 = (x + 2)(x + 3).

Correct Answer is : (x + 2) (x + 3)

Step: 1

Step: 2

Step: 3

- 1, - 24 | - 25 | 24 | |

- 3, - 8 | - 11 | 24 |

[Select the values of p and q by trial and error.]

Step: 4

The required values of p and q are - 3, - 8.

[p + q = - 11, p q = 24.]

Step: 5

Therefore, x 2 - 1 1 x + 2 4 = (x - 3)(x - 8).

[Substitute for p and q .]

Correct Answer is : (x - 3)(x - 8)

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